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A planeswalker is treated as if its text box included, "This permanent enters the battlefield with a number of loyalty counters on it equal to its printed loyalty number." This ability creates a replacement effect.

2 Answers
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A planeswalker enters the battlefield with double the normal counters, but the counters gained or lost from activating one of its abilities are unchanged.

Doubling Season doubles the loyalty counters placed on a planeswalker entering the battlefield.

You've correctly identified the relevant rules.

From the comp rules:

306.5b A planeswalker is treated as if its text box included, "This permanent
enters the battlefield with a number of loyalty counters on it equal
to its printed loyalty number." This ability creates a replacement
effect.

And Doubling Season's Oracle rulings:

10/1/2005: Doubling Season affects cards that "enter the battlefield with" a
certain number of counters. Triskelion, for example, would enter the
battlefield with six +1/+1 counters on it rather than three.

Doubling Season does not affect loyalty gain and loss from paying the costs for planeswalker abilities.

The loyalty spent or gained to activate a loyalty ability is a cost:

606.4. The cost to activate a loyalty ability of a permanent is to put on or remove from that permanent a certain number of loyalty counters,
as shown by the loyalty symbol in the ability's cost.

The game rules distinguish between costs and effects. A cost is a payment you make to perform an action:

117.1. A cost is an action or payment necessary to take another action or to stop another action from taking place. To pay a cost, a player
carries out the instructions specified by the spell, ability, or
effect that contains that cost.

An effect is the result of that action:

609.1. An effect is something that happens in the game as a result of a spell or ability. When a spell, activated ability, or triggered
ability resolves, it may create one or more one-shot or continuous
effects. Static abilities may create one or more continuous effects.
Text itself is never an effect.

Taken together, these two rules mean that paying a cost is not an effect. While cards can change costs, Doubling Season's card text specifies "effects," so loyalty abilities are not impacted.

Doubling Season does double the loyalty counters place on a planeswalker by an effect.

+1: Put a loyalty counter on Gideon, Champion of Justice for each creature target opponent controls.

This has two parts:

The "+1" is a cost and is unaffected by Doubling Season (see above).

The "Put a loyalty counter..." part is an effect and is modified by Doubling Season.

So if your opponent had three creatures and you activate Bad Gideon's +1 ability with a Doubling Season in play, Gideon would gain seven counters: +1 when activating the ability, +6 (+3 doubled) when it resolves.

Yes, No. Doubling Season does what it's errata says. If a Planeswalker would enter the battlefield with X counters, Doubling Season causes it to enter with 2x the number of counters. Loyalty abilities add counters, but these are a cost not an effect, so Doubling Season does not replace that.

Doubling Season affects cards that "enter the battlefield with" a certain number of counters. Triskelion, for example, would enter the battlefield with six +1/+1 counters on it rather than three.

The tokens and counters that Doubling Season creates are exact copies of the kind of tokens and counters that were being created in the first place.